Super McShane identity

TL;DR

This work extends McShane-type identities to the realm of super hyperbolic geometry by establishing a precise identity for the once-punctured super torus. Building on decorated super Teichmüller theory with coordinates given by $ ext{λ}$-lengths and $ ext{μ}$-invariants, the authors develop a body–soul framework and a super Markoff map on a dual Farey-type complex, enabling a control of the super length spectrum. A key contribution is the super McShane identity, which expresses 1/2 as a sum over simple closed super geodesics with terms involving the super length $ ilde{ ext{ℓ}}_oldsymbol{γ}$ and the bosonic product $W_oldsymbol{γ}$ of odd variables; the proof relies on absolute convergence, a finite-tree expansion, and a body–soul comparison that relates the super theory to the classical (body) theory. They also establish the asymptotic growth of the simple super geodesic spectrum and a bosonic–fermionic comparison theorem, providing a foundational bridge between super Teichmüller theory and asymptotic geometry of length spectra.

Abstract

The authors derive a McShane identity for once-punctured super tori. Relying upon earlier work on super Teichmüller theory by the last two-named authors, they further develop the supergeometry of these surfaces and establish asymptotic growth rate of their length spectra.

Figures (12)

• Figure 1: Flip effect on spin structures
• Figure 2: Ptolemy transformation
• Figure 3: Flip effect on spin structures for $F^1_1$.
• Figure 4: Coordinates for $S\tilde{T}(F_1^1)$
• Figure 5: Coordinates for $\tilde{T}(F_1^1)$

Theorems & Definitions (18)

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